Finite Group Databasebeta

← #11755218

Subgroups — #11755218

Order 960.

432 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#91835C211yesmaximal, characteristic, complemented
2C5×Q8×Dic3C211yesmaximal, characteristic, complemented
3C3×#4093C211yesmaximal, characteristic, complemented
4#91468C211yesmaximal, characteristic, complemented
5#91465C211yesmaximal, characteristic
6#91430C212yesmaximal, complemented
7Dic3×D20C211yesmaximal, characteristic, complemented
8#91413C212yesmaximal, complemented
9#91386C211yesmaximal, characteristic, complemented
10#91363C212yesmaximal, complemented
11#91353C212yesmaximal, complemented
12#65916—31nomaximal, Hall
13C2×C3×C5×Q8C2211yescharacteristic
14C2×C4×D15C2211yescharacteristic
15C2×C3×D20C2211yescharacteristic
16C2×#1204C2212yes—
17C2×#1202C2212yes—
18C2×D5×Dic3C2212yes—
19#6714C2212yes—
20C4⋊Dic15C2211yescharacteristic
21#6712C2212yes—
22C5×C4⋊Dic3C2212yes—
23C5×C4⋊Dic3C2211yescharacteristic
24C4×C5×Dic3C2211yescharacteristic
25C4×C5×Dic3C2212yes—
26C3×#685C2212yes—
27C3×#685C2212yes—
28C3×C4×Dic5C2211yescharacteristic
29#6669C2211yescharacteristic
30#6669C2211yescharacteristic
31#6668C2212yes—
32#6667C2211yescharacteristic
33#6667C2211yescharacteristic
34#6665C2212yes—
35#6665C2212yes—
36#5682—51nomaximal, Hall
37C4×C5×Q8—31no—
38#4093D311yescharacteristic, complemented
39#4092—31no—
40#4042—32no—
41#4041—31no—
42#4040—32no—
43#4034—31no—
44#4022—32no—
45#4021—32no—
46C4×D20—31no—
47#4018—31no—
48C22×D15C2311yescharacteristic
49C22×C3×D5C2312yes—
50C3×C5×Q8—22no—
51C2×C3×C4×C5C2312yes—
52C2×C3×C4×C5C2311yescharacteristic
53C2×Dic15C2311yescharacteristic
54C2×Dic15C2312yes—
55C4×D15—22no—
56C2×C5×Dic3C2311yescharacteristic
57C2×C5×Dic3C2312yes—
58C2×C5×Dic3C2311yescharacteristic
59C2×C3×Dic5C2311yescharacteristic
60C2×C3×Dic5C2311yescharacteristic
61C3×D20—22no—
62#1204—42no—
63#1202—42no—
64D5×Dic3—42no—
65C3×#124—51no—
66#946—51no—
67Q8×Dic3D511yescharacteristic, complemented
68#938—51no—
69D4×Dic3—51no—
70#886—52no—
71#885—52no—
72#879—52no—
73#878—52no—
74#872—51no—
75#869—51no—
76C2×C5×Q8C2×D311yescharacteristic
77C2×D20—32no—
78C2×D20C2×D311yescharacteristic
79C2×C4×D5—31no—
80C2×C4×D5—32no—
81C2×Dic10—32no—
82C5×C4⋊C4—31no—
83C5×C4⋊C4—32no—
84C42×C5—31no—
85C42×C5—32no—
86#685—32no—
87#685—32no—
88#685—32no—
89#685—31no—
90#685—31no—
91#685C2×D312yes—
92#685C2×D312yes—
93C4⋊Dic5—31no—
94#683—32no—
95#683—32no—
96#683—31no—
97#683—31no—
98C4×Dic5C2×D311yescharacteristic
99#542—151noSylow
100C22×C3×C5C2411yescharacteristic
101C2×D15—21no—
102C2×D15—21no—
103C2×D15—21no—
104C2×C3×D5—22no—
105C2×C3×D5—22no—
106C2×C3×D5—22no—
107C3×C4×C5—22no—
108C3×C4×C5#4112yes—
109Dic15—21no—
110Dic15—22no—
111C3×Dic5—21no—
112C3×Dic5—21no—
113C5×Dic3—21no—
114C5×Dic3—22no—
115C5×Dic3—21no—
116C2×C3×Q8C2×D511yescharacteristic
117C2×C3×D4—51no—
118C2×#67—52no—
119C22×Dic3—52no—
120C2×C4×D3—51no—
121C2×Dic6—52no—
122C3×C22⋊C4—52no—
123C3×C22⋊C4—52no—
124C3×C42—51no—
125#217—52no—
126#217—52no—
127(C2×D3)⋊C4—52no—
128(C2×D3)⋊C4—51no—
129(C2×D3)⋊C4—51no—
130C4⋊Dic3—51no—
131C4⋊Dic3—52no—
132C4⋊Dic3C2×D511yescharacteristic
133C4⋊Dic3C2×D512yes—
134Dic3⋊C4—52no—
135Dic3⋊C4—51no—
136Dic3⋊C4—51no—
137C4×Dic3C2×D511yescharacteristic
138C4×Dic3C2×D512yes—
139C22×D5—31no—
140C22×D5C22×D312yes—
141C5×Q8—22no—
142C2×C4×C5—31no—
143C2×C4×C5—32no—
144C2×C4×C5—31no—
145C2×C4×C5C22×D312yes—
146C2×C4×C5C22×D311yescharacteristic
147C2×Dic5—31no—
148C2×Dic5—32no—
149C2×Dic5C22×D311yescharacteristic
150C2×Dic5C22×D311yescharacteristic
151D20—22no—
152D20—122no—
153C4×D5—62no—
154C4×D5—122no—
155Dic10—122no—
156C4⋊Q8—151no—
157#126—152no—
158#124—51no—
159#124—152no—
160#123—152no—
161C22⋊Q8—152no—
162C22⋊Q8—151no—
163#121—151no—
164C4×Q8—31no—
165C4×D4—151no—
166#117—151no—
167C2×C3×C5C2×#4111yescharacteristic
168C2×C3×C5H(5,𝔽2)11yescharacteristic
169C2×C3×C52−1+411yescharacteristic
170D15—41no—
171C3×D5—42no—
172C23×C3—52no—
173C22×D3—51no—
174C3×Q8—22no—
175C3×D4—102no—
176C2×C3×C4—51no—
177C2×C3×C4—51no—
178C2×C3×C4C22×D512yes—
179C2×C3×C4C22×D511yescharacteristic
180#67—202no—
181C2×Dic3—51no—
182C2×Dic3—52no—
183C2×Dic3—202no—
184C2×Dic3C22×D511yescharacteristic
185C2×Dic3C22×D511yescharacteristic
186C2×Dic3C22×D512yes—
187C4×D3—102no—
188Dic6—202no—
189C22×C5C23×D311yescharacteristic
190C2×D5—22no—
191C2×D5—22no—
192C2×D5—22no—
193C2×D5—61no—
194C2×D5—61no—
195C2×D5—61no—
196C4×C5—22no—
197C4×C5—62no—
198C4×C5—61no—
199C4×C5—61no—
200C4×C5#23712yes—
201Dic5—21no—
202Dic5—21no—
203Dic5—61no—
204Dic5—62no—
205C2×Q8—152no—
206C2×Q8D3×D511yescharacteristic
207C2×D4—51no—
208C2×D4—152no—
209C22×C4—152no—
210C22×C4—151no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—151no—
214C4⋊C4—151no—
215C4⋊C4—152no—
216C4⋊C4—151no—
217C4⋊C4—152no—
218C22⋊C4—52no—
219C22⋊C4—52no—
220C22⋊C4—151no—
221C22⋊C4—151no—
222C22⋊C4—152no—
223C22⋊C4—152no—
224C22⋊C4—152no—
225C42—31no—
226C42—32no—
227C42—51no—
228C3×C5#54211yescharacteristic, complemented, Hall
229C22×C3—102no—
230C22×C3—102no—
231C22×C3—102no—
232C22×C3C23×D511yescharacteristic
233C2×D3—101no—
234C2×D3—101no—
235C2×D3—101no—
236C3×C4—22no—
237C3×C4—101no—
238C3×C4—101no—
239C3×C4#70912yes—
240Dic3—21no—
241Dic3—22no—
242Dic3—21no—
243Dic3—101no—
244Dic3—102no—
245C2×C5C2×#23711yescharacteristic
246C2×C5#100911yescharacteristic
247C2×C5#101011yescharacteristic
248D5—42no—
249D5—121no—
250C23—52no—
251C23—151no—
252Q8—22no—
253Q8—602no—
254D4—102no—
255D4—602no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—32no—
259C2×C4—51no—
260C2×C4—51no—
261C2×C4—151no—
262C2×C4—152no—
263C2×C4—302no—
264C2×C4—602no—
265C2×C4C2×D3×D512yes—
266C2×C4C2×D3×D511yescharacteristic
267C2×C3—202no—
268C2×C3C2×#70911yescharacteristic
269C2×C3#414711yescharacteristic
270C2×C3#414911yescharacteristic
271D3—201no—
272C5#568211yescharacteristic, complemented, Sylow
273C22—102no—
274C22—102no—
275C22—102no—
276C22—301no—
277C22—301no—
278C22—301no—
279C22C22×D3×D511yescharacteristic
280C4—22no—
281C4—61no—
282C4—61no—
283C4—62no—
284C4—101no—
285C4—101no—
286C4—302no—
287C4—301no—
288C4#677912yes—
289C3#6591611yescharacteristic, complemented, Sylow
290C2—202no—
291C2—601no—
292C2#9202811yescharacteristic
293C2#9203511yescharacteristic
294C2C2×#677911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×Q8C22
prime radical#91465C2
hypercenterC2×Q8D3×D5
hypocenterC3×C5#542
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #91835C2semidirect
2. C5×Q8×Dic3C2semidirect
3. C3×#4093C2semidirect
4. #91468C2semidirect
6. #91430C2semidirect
7. Dic3×D20C2semidirect
8. #91413C2semidirect
9. #91386C2semidirect
10. #91363C2semidirect
11. #91353C2semidirect
38. #4093D3semidirect
67. Q8×Dic3D5semidirect
228. C3×C5#542semidirect
272. C5#5682semidirect
289. C3#65916semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#918352
C5×Q8×Dic32
C3×#40932
#914682
#914652
#914302
Dic3×D202
#914132
#913862
#913632
#913532
#659163
#56825